4 Answers
4

As for the pronoun, both him and his -- respectively, the ACC-ing complementizer and the POSS-ing complementizer, as they're called in the trade -- are acceptable as the subject of the gerund complement clause.

POSS-ing is slightly more formal and more often written, and may be claimed to be "more grammatical" or "the only correct choice" or something of the sort. But it's your choice, really.

As for the parse, there are 3 clauses, because there are three verbs: want, destroy, achieve

There is, as usual, one main clause (S₀)

S₀ = He does not want S₁

with an A-Equi infinitive complement clause S₁as the direct object of want:

Is it conceivable that dream of could be parsed as a nominalized phrasal verb, as might be suggested by "I > my dream of Jeannie with the light brown hair"?
– StoneyBDec 13 '12 at 22:33

Yes, but a really deep parse like that would involve a lot more transformations. And specifications and boundary conditions for them. Jim McCawley used to call stuff like that "remnants of deceased clauses". It's like dissecting a skeleton, in the sense that things that are no longer there can't be sensed -- except from the inside, by the speaker, and everybody's different inside.
– John LawlerDec 13 '12 at 23:05

1

de mortuis nil I guess. I started on an answer, but when it got to 800 words gave up!
– StoneyBDec 13 '12 at 23:07

It's been done; it was a stock-in-trade for Generative Semantics, which is the theory I was trained in. Nowadays I'm not very doctrinaire, but I'll allow that the closer the parse is to the logical tree, the better I like it.
– John LawlerDec 13 '12 at 23:10

Is achieving always a "gerund" here, regardless of whether it's preceded by a possessive form? There seems to be some difference in exactly how it can be "modified", in that I can (just about) go along with his achieving of a Cambridge degree, but that doesn't work at all for me with him achieving of a Cambridge degree.
– FumbleFingersMar 13 '15 at 16:52

The ACC-ing structure may be more appropriate than the POSS-ing structure on occasion, and vice versa.

We watched him leaving the building to see if he remembered to lock up.

We expected his leaving the company to take place long before it actually did.

The variant with him focuses more on the person, the variant with his more on the event. In the above two examples, this strongly suggests, or dictates in the second case, which variant should be used. In the original, I think him (more personal) just shades it. Either is grammatical.

Watch is an atypical verb, like all sense verbs; note that a bare infinitive is equally good with him if watch is the verb. The second sentence is rather strange; I would expect to take place or to occur as the predicate, instead of be, and long before instead of far before.
– John LawlerDec 13 '12 at 18:56

2

Yep - the sneaking up of teatime on one doesn't encourage the formulation of well-considered examples.
– Edwin AshworthDec 13 '12 at 23:39

@EdwinAshworth That is really clear and helpful - him focuses on the person; his on the event.
– Nigel JFeb 10 '18 at 16:16

You’re on the right lines in thinking in terms of postmodification. The whole of of him achieving a Cambridge degree postmodifies the noun dream in that it answers the question ‘What kind of dream?’ The dream is about him doing something, the something being achieving a Cambridge degree.

"Him achieving" is common, but "his achieving" is grammatically preferred and simpler to explain. The preposition "of" goes with "achieving," which functions as a gerund (-ing verb used as a noun), not a participle.

You need to add a reference supporting your claim that ' "his achieving" is grammatically preferred'. I agree with John Lawler that 'both him and his ... are acceptable ...'.
– Edwin AshworthAug 13 '16 at 21:30